Math Problem Statement
Find for which values of a the following matrix is invertible:
\[
\begin{pmatrix}
2 & a - 4 & 0 \\
a & 2 & 4 \\
1 & 1 & 2
\end{pmatrix}
\]
Solution
To determine when the given matrix is invertible, we need to check when its determinant is non-zero. The matrix is:
2 & a - 4 & 0 \\ a & 2 & 4 \\ 1 & 1 & 2 \end{pmatrix}$$ We can calculate the determinant of this $$3 \times 3$$ matrix, $$\det(A)$$, as follows: $$\det(A) = 2 \cdot \begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} - (a - 4) \cdot \begin{vmatrix} a & 4 \\ 1 & 2 \end{vmatrix} + 0 \cdot \begin{vmatrix} a & 2 \\ 1 & 1 \end{vmatrix}$$ ### Step-by-Step Calculation: 1. **Calculate each minor determinant:** - For the first term, $$\begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} = 2 \cdot 2 - 4 \cdot 1 = 4 - 4 = 0$$. - For the second term, $$\begin{vmatrix} a & 4 \\ 1 & 2 \end{vmatrix} = a \cdot 2 - 4 \cdot 1 = 2a - 4$$. 2. **Expand using cofactor expansion:** $$\det(A) = 2 \cdot 0 - (a - 4) \cdot (2a - 4)$$ Simplify this expression: $$\det(A) = -(a - 4)(2a - 4)$$ 3. **Further simplification:** Expanding $$-(a - 4)(2a - 4)$$: $$\det(A) = -(2a^2 - 8a - 4a + 16) = -2a^2 + 12a - 16$$Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Determinants
Formulas
Determinant of a 3x3 matrix
Theorems
Invertibility Condition: A matrix is invertible if and only if its determinant is non-zero.
Suitable Grade Level
Undergraduate (Linear Algebra)